Royce, Boolean Rings, and the T-Relation

This paper examines in detail the four-place Boolean relation that in his article “An Extension of the Algebra of Logic” Royce calls “the T-Relation,” and the paper proves correct Royce’s major mathematical claims about the T-Relation. The paper also argues that in his article Royce develops the concept of a Boolean Ring as well as an algebra that is in essence identical to the Zhegalkin Algebra developed by the father of the Russian school of logic Ivan Ivanovich Zhegalkin. The paper finds it ironic that Royce seems to have underestimated the importance of his own work. Finally, the paper argues that there is nothing special about the T-Relation’s having four places, and it completely generalizes the T-Relation into a Boolean relation having an arbitrary number of places.